Question

Data from a poll conducted by Travelocity led to the following estimates: Approximately \(40 \%\) of travelers check their work e-mail while on vacation, about \(33 \%\) take cell phones on vacation in order to stay connected with work, and about \(25 \%\) bring laptop computers on vacation (San Luis Obispo Tribune, December 1,2005\()\). a. What additional information about the survey would you need in order to decide if it is reasonable to generalize these estimates to the population of all American adult travelers? b. Assuming that the given estimates were based on a representative sample, do you think that the estimates would more likely be closer to the actual population values if the sample size had been 100 or if the sample size had been \(500 ?\) Explain.

Short answer

To generalise these results, we need more info about the survey like sample size, selection method, etc. Larger sample sizes (500 instead of 100, for instance) increase precision, assuming the sample is representative

Step-by-step solution

Identify the needed additional information

To increase the reliability of the survey results, additional information is needed such as: 1. The sample size of the survey. 2. The method by which the participants were chosen. 3. Whether the sample was randomly chosen. 4. The location and time when the survey was conducted. 5. The demographics of the participants including age, gender, profession, etc. This information is necessary to check for any sampling bias, and to make sure that this sample is a good representative of all American adult travelers.

Analyze the effect of the sample size

Larger sample sizes generally result in estimates that are closer to the true population values. This is because they are more likely to capture the diversity and variability of the population. If the sample is representative, increasing the sample size from 100 to 500 would make the estimates more accurate and reliable.

Overall judgement

Thus, the specifics of the survey is critical for making credible generalizations about the population. And, if the sample is representative, increasing the sample size would likely yield estimates closer to the population values.

Key concepts

Sampling Bias

Sampling bias refers to a flaw in the sampling process that produces a sample that is not representative of the population. This can occur when certain members of the population are systematically more likely to be selected for the sample, or if certain portions are under-represented or excluded altogether.

For instance, if Travelocity polled individuals who booked travel through their website, the sample might over-represent tech-savvy travelers who are more inclined to use technology during vacations. To avoid sampling bias:

• Ensure the samples are selected randomly.
• Use stratified sampling to obtain a balanced representation across different subgroups.
• Avoid convenience sampling which may only capture a subset of the population that is easily accessible.

Understanding and mitigating sampling bias is crucial for obtaining reliable and valid survey results.

Representative Sample

A representative sample accurately reflects the characteristics of the population from which it is drawn. It's a mini-version of the population that includes individuals proportional to their prevalence in the overall group.

In Travelocity's survey, to check if the sample is representative, one should examine factors such as age distribution, gender, income levels, and frequency of travel. A representative sample must include a mixture of occasional and frequent travelers, individuals of varying technology-use habits, and so on, ensuring it represents all American adult travelers. Characteristics of the sample should align with known demographics of the actual population for the results to be generalized confidently.

Sample Size

Sample size is the number of participants included in a survey and it significantly influences the accuracy of the population estimates. In general, the larger the sample size, the more precise the estimate, as it reduces the margin of error and increases the level of confidence in the findings.

In the context of the Travelocity survey, increasing the sample size from 100 to 500 would likely result in estimates more reflective of the actual behaviors of American adult travelers, given that the sample is representative. This would decrease the statistical fluctuations known as sampling error, allowing for a more trustworthy extrapolation to the entire population.

Population Estimates

Population estimates are numerical values indicating the characteristics of an entire population based on the data collected from a sample. The goal is to extrapolate findings from the sample to the population, with an assumption that the trends in the sample are indicative of the population as a whole.

For credible population estimates, one must ensure the sample is free from bias and is sufficiently large. Precision of these estimates also hinges on using correct statistical methods and considering confidence intervals, which suggest the range within which the true population parameter lies, given the sample data. Assessing reliability includes determining how closely the survey questions reflect the actual behaviors or opinions being estimated, as was attempted in the case of the Travelocity data on travelers' technology use.